As shown schematically in the figure, two vessels contain water solutions (at temperature $T$ ) of potassium permanganate $\left( KMnO _4\right)$ of different concentrations $n_1$ and $n_2\left(n_1>n_2\right)$ molecules per unit volume with $\Delta n=\left(n_1-n_2\right) \ll n_1$. When they are connected by a tube of small length $\ell$ and cross-sectional area $S , KMnO _4$ starts to diffuse from the left to the right vessel through the tube. Consider the collection of molecules to behave as dilute ideal gases and the difference in their partial pressure in the two vessels causing the diffusion. The speed $v$ of the molecules is limited by the viscous force $-\beta v$ on each molecule, where $\beta$ is a constant. Neglecting all terms of the order $(\Delta n)^2$, which of the following is/are correct? ( $k_B$ is the Boltzmann constant)-
$(A)$ the force causing the molecules to move across the tube is $\Delta n k_B T S$
$(B)$ force balance implies $n_1 \beta v \ell=\Delta n k_B T$
$(C)$ total number of molecules going across the tube per sec is $\left(\frac{\Delta n}{\ell}\right)\left(\frac{k_B T}{\beta}\right) S$
$(D)$ rate of molecules getting transferred through the tube does not change with time
IIT 2020, Advanced
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$n _1>\left( n _1- n _2\right)=\Delta n$
$p _1=\frac{ n _1 RT }{ N _{ A }} \quad p _2=\frac{ n _2 RT }{ N _{ A }}$
$F =\left( n _1- n _2\right) k _{ B } T S=\Delta nk _{ B } T S(A)$
$V=\frac{\Delta n k_B T S}{\beta}$
Force balance $\Rightarrow$ Pressure $\times$ Area $=$ Total number of molecules $\times \beta v$
$\Delta k _{ B } TS =\ell n _1 S \beta v$
$\Rightarrow n _1 \beta v \ell=\Delta k _{ B } T$
Total number of molecules $/ sec =\frac{\left( n _1 vdt \right) S }{ dt }$
$= n _1 vS =\frac{\Delta k _{ B } TvS }{\beta v \ell}$
$=\left(\frac{\Delta n }{\ell}\right)\left(\frac{ k _{ B } T }{\beta}\right) S$ ($C$)
As $\Delta n$ will decrease with time therefore rate of molecules getting transfer decreases with time.
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